Wigner functions for a class of semidirect product groups

Wigner functions for a class of semidirect product groups

Abstract

We define and construct Wigner functions for the class of semidirect product groups [Special characters omitted.] whose linear part H ✹ GL ( n , [Special characters omitted.] ) is a closed Lie subgroup of GL ( n , [Special characters omitted.] ) admitting at least one open and free orbit in [Special characters omitted.] . Such groups are classified up to conjugacy in dim n = 3 and in dim n = 4 under the further requirement that they possess a semisimple ideal. The general construction is based on three main requirements: (i) the exponential map exp : [Special characters omitted.] [arrow right] G has a dense image in G , with complement of (left or right) Haar measure zero; (ii) the group admits a square-integrable representation; (iii) the Lebesgue measure d X * in the dual of the Lie algebra can be decomposed as [Special characters omitted.] , [Special characters omitted.] where [Special characters omitted.] denotes a coadjoint orbit parametrized by an index [Special characters omitted.] , [Special characters omitted.] is a measure on the parameter space, [Special characters omitted.] is a positive function on that orbit and [Special characters omitted.] is the invariant measure under the coadjoint action of G . We discuss in detail all these elements in the case of semidirect product groups [Special characters omitted.] of the kind described above and give an explicit form of the generalized Wigner function related to them. Cases of special interest are those for which the domain of the generalized Wigner function can be endowed with the structure of phase space: a sufficient condition for this to be is given in terms of purely geometrical properties of the coadjoint orbits. Relevant examples are discussed with emphasis on the case of the quaternionic group as a 4-dimensional wavelet group; this is a natural non-abelian extension of the known notions of wavelet groups in 1 and 2 dimensions which have extensive applications in signal analysis